Nonmonotonic Composition Dependence of Viscosity upon Adding Single-Chain Nanoparticles to Entangled Polymers

Well-characterized single-chain nanoparticles (SCNPs), synthesized from a linear polystyrene precursor through an intramolecular [4 + 4] thermal cycloaddition cross-linking reaction in dilute conditions, were added to entangled polystyrene melts at different concentrations. Starting from the pure linear melt, which is much more viscous than the melt of SCNPs, the zero-shear viscosity increased upon the addition of nanoparticles and reached a maximum before eventually dropping to the value of the SCNP melt. Molecular simulations reveal the origin of this unexpected behavior, which is the interplay of the very different compositional dependences of the dynamics of the two components. The SCNPs become much slower than the linear chains as their concentration decreases because they are threaded by the linear chains, reaching a maximum viscosity which is higher than that of the linear chains at a fraction of about 20%. This behavior is akin to that of single-loop ring polymers when added to linear matrices. This finding provides insights into the design and use of SCNPs as effective entropic viscosity modifiers of polymers and contributes to the discussion of the physics of loopy structures.


Synthesis and characterization of the SCNPs
The synthetic route was followed according to the literature. 1,2The synthesis of the PS-co-PVBCB linear precursors was achieved through nitroxide-mediated polymerization (NMP) to ensure the random polymerization of Styrene (S) and 4-vinylbenzocyclobutene (4-VBCB) due to different reactivity ratios.The SCNPs formation was performed in dibenzyl ether solvent in high dilution under an inert atmosphere at 250 °C.The resulting SCNPs were characterized by size exclusion chromatography (SEC) in THF, 1 H NMR (400 MHz, CDCl3), and DLS in toluene solvent.

Characterization of SCNPs and blends with linear chains: morphologies and molecular characteristics
The average number of loops per nanoparticle (  ) and the weight-average molar mass per loop ( , , ) can be calculated from equations ( 1) and (2), respectively (the molar mass of the PS repeat unit, mrep=104 gmol -1 ): Where N PS , N CrF and  ,  are the total number of PS and crosslinker monomers and the average degree of loop polymerization, respectively.
Thermal analysis was conducted with a DSC 250 (TA Instruments, USA).The samples (with weight between 3 and 5 mg) were encapsulated into Tzero aluminium pan-lid holders.Time constants and capacitances of the samples were determined based on the reference sensor with sapphire standard for heat capacity and indium standard for temperature and enthalpy calibration.
Heating and cooling rates were fixed to 10℃/min at a temperature range from 20 to 160℃.The DSC traces revealed the glass transition temperature, Tg, at the second heating scan.The DSC traces for pure SCNPs and blends with linear polystyrene chains are provided in Figure S2 below.
It should be noted that, based on the synthesis method, there should be some heterogeneity in the SCNP crosslinking density that may affect the Tg, hence the overall uncertainly in its value can be in the range 5-10%.This does not affect the present rheological results since the viscoelastic spectra are compared with respect to the same high-frequency crossover and the reported viscosities are relative (Figs. 2 and 3). is evaluated as the midpoint using TRIOS software.See also Table S1.4).Filled symbols correspond to PS matrix (filled red circle) and pure SCNPs (triangle 5%-57k, diamond 5%-140k, star 5%-330k) with different cross-liker fraction (CrF) and/or the length of the precursor copolymer.Unfilled symbols stand for blends of SCNPs with the linear matrix (triangle 5% SCNPs, triangle with cross18%, diamond 50%, star 95%).

Threading analysis and partial intramolecular coherent scattering functions
For visualization purposes only, the grey-colored linear chain is made "thicker", by spanning a triangular facet on every three consecutive beads.

Primitive path analysis and threading numbers
We performed the PPA similarly to the one in Everaers et al. 4 , where we dropped the temperature to 0.01, fixed the ends of the linear chains, removed the intrachain repulsion, but kept the interchain one.The SCNPs are left unfixed as rings in RL blends of Ref. [3], therefore they can contract to a point if not threaded.The number of entanglements Z was obtained from the linear chains only.
Although Z depends on SCNP, The curves (1-SCNP)Z(SCNP) is within errorbars the same as (1-SCNP)Z(0).For comparison, with we computed also the number of threadings Nt as in Ref. [3] from the PPA conformations: The SCNPs that have not contracted to a point were threaded.For a threading of the SCNP by a linear chain was considered a contact of the former with the latter, but only if any of the contacting monomers of the linear chain are at least contour distance Ne from the ends.Clearly this definition overestimates the threading number as there are contacts without a real topological threadings (see a snapshot in Fig. S13).

Figure S4 .
Figure S4.The signal at 3.1-3.2ppm which corresponds to the cyclobutene protons of the crosslinkers (S4α) disappears after the intramolecular crosslinking reaction which is strong evidence that all crosslinkers interfere during the thermal reaction (S4β).

Figure S6 .
Figure S6.DSC second heating curves of pure SCNPs with rate 10 K/min.Glass transition

Figure S11 .
Figure S11.Schematic illustration of threadings of distinct rings from the cycle basis.Here we show the sketch of cycles.In the present example, the cycle basis is formed by two of the three color-coded cycles (A,B,C).Note that if the basis is formed by the cycles A and B, then whenever B is threaded, A is threaded too.

Figure S12 .
Figure S12.Partial intramolecular coherent scattering functions, averaged over all the linear chains (left) or all the SCNPs (right), and considering only pairs of beads belonging to the same molecule.

Figure S13 .Figure S14 .
Figure S13.Snapshot from the system proving the overestimate of Nt: the red SCNP is threaded by blue and green linear chains.Orange linear chain is touching the SCNP too, but not threading.Other chains are not shown.

Table S1 .
Compositions and respective Tg values of the SCNPs-linear PS blends used.